# High-Order Block Support Spatial Simulation Method and Its Application at a Gold Deposit

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## Abstract

High-order sequential simulation methods have been developed as an alternative to existing frameworks to facilitate the modeling of the spatial complexity of non-Gaussian spatially distributed variables of interest. These high-order simulation approaches address the modeling of the curvilinear features and spatial connectivity of extreme values that are common in mineral deposits, petroleum reservoirs, water aquifers, and other geological phenomena. This paper presents a new high-order simulation method that generates realizations directly at the block support scale conditioned to the available data at point support scale. In the context of sequential high-order simulation, the method estimates, at each block location, the cross-support joint probability density function using Legendre-like splines as the set of basis functions needed. The proposed method adds previously simulated blocks to the set of conditioning data, which initially contains the available data at point support scale. A spatial template is defined by the configuration of the block to be simulated and related conditioning values at both support scales, and is used to infer additional high-order statistics from a training image. Testing of the proposed method with an exhaustive dataset shows that simulated realizations reproduce major structures and high-order relations of data. The practical intricacies of the proposed method are demonstrated in an application at a gold deposit.

## Keywords

Sequential high-order simulation Block support Cross-support joint probability density function High-order spatial statistics## 1 Introduction

Stochastic simulation methods are used to quantify the spatial uncertainty and variability of pertinent attributes of natural phenomena in geosciences and geoengineering. Initial simulation methods were based on Gaussian assumptions and second-order statistics of corresponding random field models (Journel and Huijbregts 1978; David 1988; Goovaerts 1997). To address the limits of such Gaussian approaches, multiple point statistics (MPS)-based simulation methods were introduced (Guardiano and Srivastava 1993; Strebelle 2002; Zhang et al. 2006; Arpat and Caers 2007; Remy et al. 2009; Mariethoz et al. 2010; Mariethoz and Caers 2014; Mustapha et al. 2014; Chatterjee et al. 2016; Li et al. 2016; Zhang et al. 2017) to remove distributional assumptions, as well as to enable the reproduction of complex curvilinear and other geologic features by replacing the random field model with a framework built upon extraction of multiple point patterns from a training image (TI) or geological analogue. The main limitations of MPS methods are that they do not explicitly account for high-order statistics, nor do they provide consistent mathematical models as they generate TI-driven realizations. Previous studies have shown resulting realizations that comply with the TI used but do not necessarily reproduce the spatial statistics inferred from the data (Osterholt and Dimitrakopoulos 2007; Goodfellow et al. 2012). As an alternative, to address the above limitations, a high-order simulation (HOSIM) framework has been proposed as a natural generalization of the second-order-based random field paradigm (Dimitrakopoulos et al. 2010; Mustapha and Dimitrakopoulos 2010a, b, 2011; Minniakhmetov and Dimitrakopoulos 2017a, b; Minniakhmetov et al. 2018; Yao et al. 2018). The HOSIM framework does not make any assumptions about the data distribution, and the resulting realizations reproduce the high-order spatial statistics of the data. Similar to the MPS and most Gaussian simulation approaches, HOSIM methods generate realizations at the point support scale, whereas in most major areas of application, simulated realizations must be at the block support scale. Typically, the change of support scale needed is addressed by generating simulated realizations on a very dense grid of nodes that is then postprocessed to generate realizations at the block support size needed. This is a computationally demanding process, as related configurations may require extremely dense grids with on the order of many millions to billions of nodes. Thus, there is a need for computationally efficient methods that simulate directly at the block support scale.

In the context of conventional second-order geostatistics, direct block support simulation has been proposed. An approach termed “direct block simulation” was presented by Godoy (2003), which discretizes each block into several internal nodes, but only stores a single block value in memory for the next group simulation. This mechanism drastically reduces the amount of data stored in memory and saves considerable computational effort. The sequential direct block simulation method was expanded by Boucher and Dimitrakopoulos (2009) to incorporate multiple correlated variables by applying min/max autocorrelation factors. An explicit change of the support model and direct simulation at block support scale were used by Emery (2009). Although efficient, these methods carry all the limitations of a Gaussian simulation framework, and the related spatial connectivity is limited to two-point spatial statistics, thus they remain unable to characterize non-Gaussian variables, complex nonlinear geological geometries, and the critically important connectivity of extreme values (Journel 2018). Alternatives are, therefore, needed.

High-order sequential simulation methods use high-order spatial cumulants to describe complex geologic configurations and high-order connectivity. At the same time, simulated realizations remain consistent with respect to the statistics of the available data, while capitalizing on the additional information that TIs can provide. These high-order spatial cumulants are described by Dimitrakopoulos et al. (2010) as combinations of moment statistical parameters. A high-order simulation algorithm was proposed by Mustapha and Dimitrakopoulos (2010a), where the conditional probability density functions (cpdf) are approximated by Legendre polynomials and high-order spatial cumulants. A template is defined based on the central node to be simulated and the nearest conditioning data. The replicates of this configuration are obtained from both the data and TI, and are used as input for the calculation of the Legendre coefficients in the cpdf approximation. Advantages of this method lie in the absence of assumptions on the distribution of the data and in being a data-driven approach. The Legendre polynomial was replaced by Legendre-like splines as the basis function for the estimation of conditional probabilities by Minniakhmetov et al. (2018). Results show a more stable approximation of the related cpdf. Improving upon the computational performance, Yao et al. (2018) proposed a new approach, where the calculation of the cpdf is simplified and no explicit calculation of cumulants is required. Although effective, the methods described above are performed at point support scale.

This paper presents a new method that generates high-order stochastic simulations directly at the block support scale. The technique considers overlapping grids representing a study area at two support scales, viz. point and block, where the simulation process is implemented at the latter. In the sequential simulation process followed, only the initial point support data and previously simulated blocks are added to the set of conditioning values, thus drastically reducing the number of elements stored in memory. The block to be simulated and the nearest conditioning data, at the point or block support scale, define the spatial configuration of the template used. Similarly, the TI is represented at both support scales to provide replicates of related spatial template configurations. The conditional cross-support joint density function estimated at each block is approximated by Legendre-like splines.

The remainder of the paper is organized as follows: First, the proposed model for high-order block support simulation is presented. Subsequently, a case study in a controlled environment assesses the performance of the current approach. Next, the method is applied to an actual gold deposit to demonstrate its practical aspects. Conclusions follow.

## 2 High-Order Block Support Simulation

### 2.1 Sequential Simulation

*j*in the domain \( D \subseteq R^{n} . \) Now, consider a transformation function that takes the above point support RF to the block support RF. Any upscaling function can be applied, but assume Eq. (1) for simplicity

*V*is the volume.

### 2.2 Joint Probability Density Function Approximation

Now, to determine \( L_{i \ldots jk \ldots l} \), the expected value from Eq. (10) is calculated from replicates of the training image according to a template defined from the simulation grid and sampling data.

- 1.
Upscale the TI from point to block support scale.

- 2.
Define a random path to visit all the unsampled block locations on the simulation grid.

- 3.At each \( v^{0} \) block location:
- (a)
Find the nearest conditioning point and block support values.

- (b)
Obtain the template \( \tau \) according to the configuration of the central block and related conditioning values at both support scales.

- (c)
Scan the training images, searching for replicates of the template \( \tau \) and corresponding values.

- (d)
Calculate all the spatial cross-support coefficients \( L_{i \ldots jk \ldots l} \) using Eq. (10).

- (e)
Derive the conditional cross-support jpdf \( f_{{Z_{0}^{V} }} \left( {z_{0}^{V} \left| {d_{P} ,z_{1}^{V} ,z_{2}^{V} , \ldots ,z_{k}^{V} } \right.} \right) \) according to Eqs. (8) and (3).

- (f)
Draw a uniform value from \( \left[ {0,1} \right] \) to sample \( z_{0}^{V} \) from the conditional cumulative distribution derived from the above.

- (g)
Add \( z_{0}^{V} \) to the simulation grid at block support scale so that it can be a conditioning value for the next block.

- (a)
- 4.
Repeat steps 2 and 3 for additional realizations.

### 2.3 Approximation of a Joint Probability Density Using Legendre-Like Orthogonal Splines

## 3 Testing with an Exhaustive Dataset

*U*and

*V*with sizes of 260 × 300 pixels. Random stratified sampling is used to retrieve 234 values from or 0.3 % of the exhaustive image

*V*to be used as the dataset in the direct block simulation of

*V*, to test the proposed method. The full dataset

*V*is converted from the point to a block support representation by averaging over 5 × 5 pixels. This block support version is referred to here as the fully known reference image and is used for comparisons. Figure 2 shows

*V*at the point and block support scale, as well as the dataset to be used. The image

*U*is chosen as the training image in the simulation process. Figure 3 presents the TI at both point and block support (5 × 5 unit size) scales. To help the method find more meaningful spatial patterns of the potential conditioning templates, the histogram of the TI is matched to that of the dataset. Histograms of the exhaustively known image, TI, and dataset are displayed in Fig. 4, and basic statistics are presented in Table 1.

Basic statistics of dataset, training image, and fully known image at point support scale

Basic statistic | Dataset | Reference image | Training image |
---|---|---|---|

Average | 277.0 | 278 | 278.5 |

Median | 193.5 | 221.3 | 193.6 |

Variance | 68,926 | 62,423 | 70,385 |

*V*dataset at block support scale, using the data and the training image mentioned above. Note that the maximum number of knots used (Eq. 12) is 50, which provides computationally efficient and stable polynomial approximations, as appropriate. Figure 5 shows three of the simulated realizations generated, and Table 2 presents the statistics related to the average of the 15 simulations, training image, and reference image at block scale. Comparison of Figs. 2b and 5 suggests that the simulations reproduce the main structures of low and high values of the fully known reference image

*V*. The histograms and variograms presented in Figs. 6 and 7 reasonably follow the behavior exhibited by the variogram model from the data and training image, respectively. Note that the variograms of the data are computed at point scale and rescaled to represent the corresponding volume–variance relation (Journel and Huijbregts 1978).

Basic statistics of the average of the simulations, training image, and reference image at block support scale

Basic statistic | Simulations | Training image | Reference image |
---|---|---|---|

Average | 280.6 | 278.5 | 278.0 |

Median | 234.6 | 237.0 | 235.4 |

Variance | 51,764 | 46,518 | 52,304 |

The fourth-order cumulant map reproduces the characteristics that are closer to the TI than the fully known image, as expected. Note that, by explicitly calculating the spatial high-order cumulants, the information received from the training image to infer local cross-support distributions is conditioned to the data.

## 4 Application at a Gold Deposit

^{2}configuration, covering an area of 4.5 km

^{2}. The training image is defined on 405 × 445 × 43 grid blocks of size 5 × 5 × 10 m

^{3}and is based on blasthole samples. Both inputs are composited in a 10 m bench and are considered to be at point support scale. Figure 10 presents the drillholes available and the training image at block scale. The deposit to be simulated is represented by 510,800 blocks, each measuring 10 × 10 × 10 m

^{3}.

Basic statistics of the average of the simulations and training image at block support scale and dataset at point scale

Basic statistic | Simulations | TI block support | Data point support |
---|---|---|---|

Average | 0.63 | 0.62 | 0.63 |

Median | 0.40 | 0.39 | 0.39 |

Variance | 0.74 | 0.76 | 1.81 |

Further highlighting the advantages of the proposed direct block high-order simulation method, note that, for this case study, the runtime of the related algorithm was approximately 5.5 h, while the point high-order simulation requires approximately 24 h. Both approaches are tested with the same specifications and computing equipment: Intel^{®} Core™ i7-7700 CPU with 3.60 GHz, 16 GB of RAM, running under Windows 7.

## 5 Conclusions

This paper presents a new high-order simulation method that simulates directly at block support scale by estimating, at every block location, the cross-support joint probability density function. Legendre-like splines are the set of basis functions used to approximate the above density function. The related coefficients are calculated from replicates of a spatial template employed. The latter template is generated from the configuration of the block to be simulated and associated conditioning values, whose support can be at both point and block scale. The high-order character of the proposed direct block simulation method ensures that the generated realizations reflect the complex, nonlinear spatial characteristics of the variables being simulated and reproduce the connectivity of extreme values.

The proposed algorithm is tested using an exhaustive image, showing that the different realizations generated can reasonably reproduce spatial architectures observed in the exhaustive image. An application at a gold deposit shows the practical aspects of the method. In addition, it documents that the method works well, while simulated realizations are shown to reproduce the spatial statistics of the available data up to the cumulants of fourth order that were calculated. Further work will focus on improving the computational efficiency, generating training images that are consistent with the high-order relations in the available data, and extending the proposed method to jointly simulate multiple variables.

## Notes

### Acknowledgements

This work is funded by the National Science and Engineering Research Council of Canada, Natural Science and Engineering Research Council of Canada (NSERC) CRD Grant CRDPJ 500414-16, the COSMO mining industry consortium (AngloGold Ashanti, Barrick Gold, BHP, De Beers, IAMGOLD, Kinross, Newmont Mining and Vale) NSERC Discovery Grant 239019, and the IAMG by the 2017 Mathematical Geosciences Student Award.

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